Rational curves with a maximum number of nodes

Counting Nodes on Rational Plane Curves

In this paper, we consider polynomial parametrized curves in the affine plane k over an algebraically closed field k. Such curves are given by κ : k → k : t 7→ (x(t), y(t)), and may or may not contain singular points. The problem of how many singular points there are is of specific importance to the theory of polynomial knots, as it gives a bound on the degrees necessary to achieve a parametriz...

The Maximum or Minimum Number of Rational Points on Curves of Genus Three over Finite Fields

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.

On the maximum number of rational points on singular curves over finite fields

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.

Tsallis Maximum Entropy Lorenz Curves

In this paper, at first we derive a family of maximum Tsallis entropy distributions under optional side conditions on the mean income and the Gini index. Furthermore, corresponding with these distributions a family of Lorenz curves compatible with the optional side conditions is generated. Meanwhile, we show that our results reduce to Shannon entropy as $beta$ tends to one. Finally, by using ac...

The Maximum Number of Edges in a Strongly Multiplicative Graph